Freudenthal theorem and spherical classes in $H_*QS^0$

TL;DR

This paper investigates spherical classes in the homology of infinite loop spaces, applying Freudenthal's theorem to establish vanishing results for certain elements and confirming a conjecture on spherical classes in specific finite loop spaces.

Contribution

It introduces a new vanishing theorem for the Hurewicz images of stable homotopy elements factoring through finite spectra and determines spherical classes in homology of certain loop spaces, confirming Eccles' conjecture.

Findings

Vanishing of Hurewicz images for elements factoring through finite spectra.

Trivial Hurewicz images for Mahowaldean families in p-local and p-complete settings.

Complete determination of spherical classes in homology of certain loop spaces.

Abstract

This note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z},\mathbb{Z}/p$ with a special focus on the case of $p=2$ related to Curtis conjecture. We apply Freudenthal theorem to prove a vanishing result for the Hurewicz image of elements in ${\pi_*^s}$ that factor through certain finite spectra. Either in $p$-local or $p$-complete settings, this immediately implies that elements of well known infinite families in ${_p\pi_*^s}$, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism ${_p\pi_*^s}\simeq{_p\pi_*}QS^0\to H_*(QS^0;\mathbb{Z}/p)$. We also observe that the image of the integral unstable Hurewicz homomorphism $\pi_*^s\simeq\pi_*QS^0\to H_*(QS^0;\mathbb{Z})$ when restricted to the submodule of decomposable elements, is given by $\mathbb{Z}\{h(\eta^2),h(\nu^2),h(\sigma^2)\}$. We apply this latter to completely determine spherical classes in $H_*(\Omega^dS^{n+d};\mathbb{Z}/2)$ for certain values of $n>0$ and $d>0$; this verifies a Eccles' conjecture on spherical classes in $H_*QS^n$, $n>0$, on finite loop spaces associated to spheres.